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Ambient spaces for cones generated by matricial inner products. (English) Zbl 1083.15034

This paper provides a short expository survey (without proofs) of the characterizations and properties of four ambient spaces for cones generated by matricial inner products on the space \({\mathcal M}_n\) of complex \(n\times n\) matrices. Results about the spaces of Hermitian-preserving transformations, real-preserving transformations, selfadjoint transformations, and the space of all transformations \(T\) such that \(\langle T(A),A\rangle\) is Hermitian for all matrices \(A\in {\mathcal M}_n\) are examined.

MSC:

15B48 Positive matrices and their generalizations; cones of matrices
15B57 Hermitian, skew-Hermitian, and related matrices
15-02 Research exposition (monographs, survey articles) pertaining to linear algebra
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